Let \(M\)  denote the Hardy–Littlewood maximal function. The Muckenhoupt–Wheeden Conjecture stems from vector-valued estimates for the maximal function. In 1971, C. Fefferman and E. Stein [1]} showed that
\(\Vert Mf\Vert _{L^{1,\infty }(w)} \lesssim \int _{\mathbb {R}^n} |f| M w (x)dx, \) 
